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40                       ANALYTICAL MECHANICS
universal law applicable to all bodies under all conditions; therefore it is applicable to rigid bodies as well as to single particles. But since rigid bodies may be subject to two distinct types of action the law may be stated in the following form.
The sum of all the linear and angular actions to which a body or a part of body is subject at any instant vanishes:
Aj ()-                                 (A')
But since the two types of action are independent of each other the sum of each type must vanish when the combined sum vanishes. Therefore we can split, the law into the following two sections.
To every linear action there is an equal and opposite linear reaction, or, the sum of all the linear actions to which a body or a part of body is subject at any instant vanishes:
SA, = 0.                                      (A/)
To every angular action there is an equal and opposite angular reaction, or, the sum of all the angular actions to which a body or a part of body is subject at any instant vanishes:
SAa = 0.                                   (Aj
50. Conditions of Equilibrium of a Rigid Body, I f we re] >lnee the term "linear action" in the first section of the law by the word "force" and the term "angular action " in the second section of the law by the word "torque" we obtain the two conditions which must be satisfied in order that a rigid body be in equilibrium. Thus, in order that a rigid body be in equilibrium the following conditions must be satisfied.
First. The sum of all the forces acting upon the rnjid bndy must vanish, that is, if F1; F2, . . . Fri denote all the forces acting upon the body then the vector equation
Fi+F2+  -  + Fn=0                      (III)
must be satisfied.